prototiles) in Euclidean n-space, En, for n > 2. The prototiles are usually required to be rather nice topologically, at least homeomorphs of the closed unit ball. One then makes arbitrarily many congruent copies, called tiles, these prototiles, and considers all ways (called tilings) that such tiles may provide a simultaneous covering packing En; tiling is thus an unordered collection which union but interiors each pair do not intersect. Wang's original problem was determine if it possible...

This article contains introductions to three open problems of significant research interest, taken from number theory, logic, and condensed matter physics.All will be shown have at their core special cases one simply-stated optimization problem.Our goal is use the intuition gained these perspectives direct attention this common core, which constitutes, in fact, problem remarkable depth importance.We also show that some tools developed separate are real value others.Since each uses jargon...

We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing first order transition curve ending in critical point.

We study a mean field model of complex network, focusing on edge and triangle densities. Our first result is the derivation variational characterization entropy density, compatible with infinite node limit. then determine optimizing graphs for small density range though we can only prove they are local, not global, maxima density. With this assumption that resulting must lose its analyticity in various regimes. In particular implies existence phase transition between distinct heterogeneous...

We show, through local estimates and simulation, that if one constrains simple graphs by their densities $\varepsilon$ of edges $\tau$ triangles, then asymptotically (in the number vertices) for over $95\%$ possible range those there is a well-defined typical graph, it has very structure: vertices are decomposed into two subsets $V_1$ $V_2$ fixed relative size $c$ $1-c$, probabilities edges, $g_{jk}$, between $v_j\in V_j$, $v_k\in V_k$. Furthermore four parameters $c, g_{11}, g_{22}$...

We present an experiment on crystallization of packings macroscopic granular spheres. This system is often considered to be a model for thermally driven atomic or colloidal systems. Cyclically shearing packing frictional spheres, we observe first order phase transition from disordered ordered state. The state consists crystallites mixed fcc and hcp symmetry that coexist with the amorphous bulk. transition, initiated by homogeneous nucleation, overcomes barrier at 64.5% volume fraction....

This is a status report on the classical problem of determining origins crystalline symmetry in low temperature matter.

We find that a column of glass beads exhibits well-defined transition between two phases differ in their resistance to shear. Pulses fluidization are used prepare static states with particle volume fractions $ϕ$ the range 0.57-0.63. The shear is determined by slowly inserting rod into beads. occurs at $ϕ=0.60$ for speeds rod.

Based on numerical simulation and local stability analysis we describe the structure of phase space edge/triangle model random graphs. We support evidence with mathematical proof continuity discontinuity for many transitions. All but one themany transitions in this break some form symmetry, use to explore how changes symmetry are related discontinuities at these

We study the asymptotics of large simple graphs directly constrained by limiting subgraph densities edges and an arbitrary fixed graph |$H$|⁠. prove that, for all but finitely many values edge density, if density |$H$| is to be slightly higher than that corresponding Erdős–Rényi graph, typical bipodal with parameters varying analytically densities. Asymptotically, depend only on degree sequence

The time evolution of a class generalized quantum Ising models (with various long-range interactions, including Dyson's 1/rα) has been studied from the C*-algebraic point view. We establish that: (1) All 〈A〉t are weakly almost periodic in time; (2) there exists unique averaging procedure over (3) thermodynamical limit can be locally implemented by effective Hamiltonians algebra quasilocal observables; (4) specific connection between spectral properties initial state and approach to...

We measure shear response in packings of glass beads by pulling a thin, rough, metal plate vertically through bed volume fraction phi, which is set, before the pulled, range 0.575 to 0.628. The yield stress velocity independent over 4 decades and increases exponentially with transition at phi approximately 0.595. An analysis measured force fluctuations indicates that modulus significantly smaller than bulk modulus.

We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, examine the consequences of different choices for definition isomorphism. In particular, we discuss role a choice plays with regard to matching rules structures.

The hard sphere model is known to show a liquid-solid phase transition, with the solid expected be either face centered cubic or hexagonal close packed. differences in free energy of two structures are very small and various attempts have been made determine which structure more stable. We contrast different approaches extend one.